Energy operator

In quantum mechanics, energy is defined in terms of the energy operator, acting on the wavefunction of the system.

Definition

It is given by:

It acts on the wavefunction (the probability amplitude for different configurations of the system)

Application

The energy operator corresponds to the full energy of a system. The Schrödinger equation describes the space- and time-dependence of slow changing (non-relativistic) wavefunction of quantum systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta.

Schrödinger equation

Using the classical equation for conservation of energy of a particle:

where E = total energy, H = hamiltonian, T = kinetic energy and V = potential energy of the particle, substituting the energy and Hamiltonian operators and multiplying by the wavefunction obtains the Schrödinger equation

that is

where i is the imaginary unit, ħ is the reduced Planck constant, and is the Hamiltonian operator.